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A Bernstein-Chernoff deviation inequality, and
geometric properties of random families of
operators
Shiri Artstein-Avidan ∗,
Mathematics Department, Princeton University
Abstract: In this paper we first describe a new deviation inequality for sums
of independent random variables which uses the precise constants appearing
in the tails of their distributions, and can reflect in full their concentration
properties. In the proof we make use of Chernoff’s bounds. We then apply
this inequality to prove a global diameter reduction theorem for abstract fam-
ilies of linear operators endowed with a probability measure satisfying some
condition. Next we give a local diameter reduction theorem for abstract fam-
ilies of linear operators. We discuss some examples and give one more global
result in the reverse direction, and exensions.
Acknowledgement: I would like to thank Prof. Vitali Milman for his support
and encouragement, and mainly for his mathematical help and advice.
∗This research was partially supported by BSF grant 2002-006.
1
The first theorem in this note is a new Bernstein-type deviation inequality
which we prove using Chernoff’s bounds. This theorem is different from the
classical Bernstein inequality in the following way: whereas the condition in
the standard Bernstein inequality is on the global behavior of the random
variables in question, for example a condition on the expectation of ecX2,
in Theorem 1 below the condition uses only the constants appearing in the
tail of the distribution, and so can reflect concentration. Sometimes one can
prove very strong estimates on the tails. In the theorem below these estimates
can be then used and are amplified when one averages many i.i.d. copies of
the variable. This theorem in a special case was brought forward and used
in the paper [AFM] for a specific example. Its proof is straightforward using
only Chernoff’s bounds, and we find this approach insightful and new.
We first apply the deviation inequality for some geometric question. We
present several results regarding the behavior of the diameter of a convex
body under some random operations. The first is a global result, namely
regarding the Minkowski sums of copies of a convex body acted upon by
abstract families of linear operators endowed with a probability measure.
The classical global diameter reduction is the well known special case where
the family of operators is O(n), the family of orthogonal rotations. This
was first observed in [BLM], see also [MiS] for more details. In Section 5 we
revisit this case as an example.
The second result we discuss is of a local nature, and is an extension of
the now well known diameter reduction phenomenon for random orthogonal
projections. This phenomenon was first observed by Milman in his proof for
the quotient of a subspace theorem, [Mi2] (and analyzed as a separate propo-
2
sition in [Mi3], where more references can be found). It can be considered
today as a consequence of the classical Dvoretzky-type theorem as proved in
[Mi1]. The classical theorem concerns the case where the random operation is
intersection with a random subspace or projection onto a random subspace.
However, in this paper we consider a more general setting. Instead of working
with projections, we deal with an abstract family of linear operators endowed
with a probability measure and find a condition on this measure (which is in
fact a condition on the probabilistic behavior of the operators on individual
elements x ∈ Rn) which promises that a diameter reduction theorem holds.
The proof of the theorem uses Talagrand’s Majorizing Measures Theorem,
see [Tal].
In Section 4 we describe a global result in the reverse direction, describing
in a particular case when does the resulting body contain a euclidean ball.
In the classical setting this kind of containment is the only known reason for
stabilization of the diameter.
We then discuss some examples. We show how the abstract propositions
indeed imply Milman’s diameter reduction theorem for usual orthogonal pro-
jections and global Dvoretzky’s Theorem for unitary transformations (and
the diameter reduction which occurs until stabilization). We describe other
families of operators for which there is a similar diameter reduction. One of
our main goals is to crystalize which properties of the operators are impor-
tant for diameter reduction results to hold. Finally we give two more variants
of the local result.
We remark that the results described in this paper have many similar vari-
ants that can be proven in exactly the same way. The choice of conditions in
3
each one depend very much on the applications in mind. Thus as much as we
tried to give general and abstract constructions, stating each proposition in
full generality would be notationally very inconvenient. We tried to indicate
in remarks which main variants are possible for each statement.
Recently I learned that results in the spirit of Proposition 3 below are be-
ing studied by the team of A. Litvak, A. Pajor and N. Tomczak-Jaegermann,
see [LPT].
Notation: We use | · | to denote the euclidan norm in Rn, and denote
by Dn the euclidean unit ball, Dn = {x : |x| ≤ 1}. For a centrally sym-
metric convex body K ⊂ Rn we denote by d = d(K) its diameter, so
K ⊂ d(K)Dn. We let M∗ = M∗(K) denote half its mean width, that is
M∗(K) =∫
Sn−1 supy∈K〈x, y〉dσ(x) where Sn−1 is the euclidean unit sphere
and σ denotes the normalized Lebesgue measure on this sphere. Thus M∗ is
the average of the dual norm of K, which we denote by ‖x‖∗ = supy∈K〈x, y〉.
1 A Deviation Inequality
We first describe our main tool, which is a Bernstein-type deviation Theo-
rem. Its proof follows from Chernoff’s bounds, and we provide it below. We
wish to point out the main difference between this theorem and the classical
Bernstein deviation inequality for, say ψ2, random variables. The classical
theorem, for which we refer the reader to, say, [BLM], gives an upper bound
for the probability in (1) below, in the following form: If A is the ψ2-norm
of the random variable X, and Xi are i.i.d. copies of X, then
P[| 1
N
N∑i=1
Xi − EX| > t] ≤ 2e−Nt2/(8A2).
4
The ψ2-norm of the variable is affected by the constant in the tail estimate,
but not only, and for example the expectation or variance may take a part
and influence this constant A. The purpose of the deviation inequality in
our Theorem 1 is to use the tail estimate itself (and not just the good ψp
behavior following from it). This type of Proposition was first used, for a
special example, in [AFM].
Theorem 1 Assume X is a random variable satisfying
P[X > t] ≤ e−Ktp
for some constant K > 0, some p > 1, and any t > K0. Let X1, . . . , XN be
i.i.d. copies of X. Then for any s > max{C(K, p), K0},
P[1
N
N∑i=1
Xi > 3s] ≤ C0e−N(Ksp−ln 2), (1)
where C0 is a universal constant for p bounded away from 1, and where
C(K, p) = (1+ln 2)
K1/p .
Remark 1. As will be evident from the proof, it is not necessary that
the variables be identically distributed, and it is sufficient that they are
independent and that each satisfies the tail estimate.
Remark 2. The term ln 2 appearing in the estimate is avoidable, by using
the exact form of Chernoff’s inequality in the proof, namely using that for
i.i.d. p-Bernoulli variables Zi, and for β < p,
P[N∑
i=1
Zi ≤ βN ] ≤ e−N [β ln(β/p)+(1−β) ln((1−β)/(1−p))].
For reference on this estimate and on the Chernoff bound used in the proof see
for example the survey on geometric applications of Chernoff type estimates
5
[AFM]. More precisely, if one substitutes the constant 3 by C1 then instead
of ln 2 one can put a constant c2 such that c2 → 0 when C1 →∞.
Remark 3. In the case p = 1 one encounters a problem with the convergence
of the probability. However, if one assumes an upper bound d on the random
variable X, then the same proof as below will give an upper estimate on the
probability in (1) of the form ≈ C0 log(d/s)e−NKs/ log(d/s), which is sufficient
in some cases.
Proof of Theorem 1. We will use the standard Chernoff bound. For j =
log s+ 1, log s+ 2, . . . we define
Aj = {2j−1 < X ≤ 2j},
so that P [Xi ∈ Aj] ≤ e−K2(j−1)p(where we have used the assumption s > K0).
We set mj = Ns2−j/(j− log s)2. We measure the probability of the following
event: out of the N variables Xi, for every j, no more than mj of them are
in Aj. This event is included in the event that
1
N
N∑i=1
Xi ≤ s(1 +∞∑
j=1
1
j2) ≤ 3s
We will estimate the probability of the complementary event. It is less
than the sum over j over the individual probabilities
Pj = P[ more than mj of the Xis are in Aj].
As long as
s2−j/(j − log s)2 > e−K2p(j−1)
(2)
6
(which will give us a condition on s, namely a lower bound on s in terms of
K and p), this probability is small, and by Chernoff it is smaller than
e−N [K2p(j−1)s2−j/(j−log s)2−ln 2] = e−N [Ks2(p−1)j−1/(j−log s)2−ln 2].
(Here by using the exact form of Chernoff’s estimate we may substitute
− ln 2 by the term β ln β + (1 − β) ln(1 − β) for, say, β = s2−j, and this
will improve the estimate. More precisely, if we sum to begin with for j =
log(C1s) + 1, log(C1s) + 2, . . . we will have above that β ≤ 1/(2C1) and so
the additional term in the exponent is also small, and tends to 0 when C1
increases.)
The sum of these probabilities converges (we are using the fact p > 1),
and is comparable to the first element in the series, which is e−N [spK/2−ln 2]
(so, in fact, C0 in the Theorem depends on p but can be taken universal, and
even not large at all, when p is bounded away from 1).
We now have to check condition (2). The left hand side is 2−i/i2 and the
right hand side is e−K2p(i−1)sp. Taking the natural logarithm of both sides we
see that the condition is (i+ 2 ln i)/2p(i−1) < Ksp. Clearly the left hand side
is largest for i = 1, 2, so we need only ensure that s > (2+2 ln 2)
2K1/p = (1+ln 2)
K1/p .
Thus we have shown that the condition in Chernoff’s bound is satisfied and
the proof of Theorem 1 is complete. ¤
2 A Global Proposition
We now state the application of this theorem, which is a global proposition
regarding the decrease of diameter of a convex body, which generalizes the
7
well known diameter reduction for averages of random orthogonal rotations
of a convex body.
Proposition 2 There exist universal constants c, C such that: Let {A} be
some family of operators A : Rn → Rn with some probability measure P.
Assume that for some body K ⊂ Rn and some α > 0 the following holds: for
every x ∈ Rn and for any s > s0
P[A : ‖Ax‖K > s|x|] ≤ e−α2s2n. (3)
Then, if T < 2( 1αs0
)2, we have with probability greater than 1− e−cn that for
any x ∈ Rn
1
T
T∑i=1
‖Aix‖K ≤ C1
α√T|x|, (4)
and if T ≥ 2( 1αs0
)2 we have with probability greater than 1−e−cn that for any
x ∈ Rn
1
T
T∑i=1
‖Aix‖K ≤ Cs0|x|, (5)
where Ai are chosen independently according to the distribution P.
Restating the proposition in geometric form, using duality, gives
Corollary 1 There exist universal constants c, C such that: Let {A} be some
family of operators A : Rn → Rn with some probability measure P. Assume
that for some body K ⊂ Rn and some α > 0 the following holds: for every
x ∈ Rn and for any s > s0
P[A : ‖A∗x‖∗K > s|x|] ≤ e−α2s2n. (6)
8
Then, if T < 2( 1αs0
)2, we have with probability greater than 1− e−cn that
A1K + A2K + · · ·+ ATK
T⊂ C
1
α√TDn, (7)
and if T ≥ 2( 1αs0
)2 we have with probability greater than 1− e−cn that
A1K + A2K + · · ·+ ATK
T⊂ Cs0Dn, (8)
where Ai are chosen independently according to the distribution P.
Remark. We are describing the case p = 2 because it is the most useful.
However, for any p > 1 we have a similar result, namely if instead of (6) we
have an estimate of the form
P[A : ‖A∗x‖∗K > s|x|] ≤ e−αpspn,
then we get a separation into two cases T < 2( 1αs0
)p and T ≥ 2( 1αs0
)p. In the
first case instead of (7) we get that the average is included in C 1αT 1/pDn, and
in the second case we get exactly (8). The proof is identical. As for p = 1,
this is different, since Theorem 1 is different, and one gets an upper estimate
on the diameter of the form ≈ 1αT
log d if this quantity is greater than s0.
The meaning of the separation of the two cases inside the proposition
seems to be that there is a diameter reduction of order√T when taking an
average of T copies of a convex body K operated upon by random operators
Ai, until the diameter reaches some critical value on which it stabilizes. In
the case of orthogonal rotations we know the reason for stabilization, namely
the body becomes a euclidean ball. Of course, the proposition above gives
only an upper bound, and by no means implies stabilization. To get any
result in the reverse direction (namely, the inclusion of a ball of some radius
9
after a given number of steps, and stabilization) we would need a reverse
condition as well, promising that points do not shrink very much under the
random operation. This is addressed in Proposition 5.
Proof of Proposition 2 We begin with the case of T < 2( 1αs0
)2, and we should
show that for any x ∈ Sn−1
T∑i=1
‖Aix‖ ≤ C
√T
α, (9)
under the assumption that for some constant α and for any s > s0
P[A : ‖Ax‖ > s|x|] ≤ e−α2s2n.
Theorem 1 with p = 2 tells us that then for s > max{s0, (1 + ln 2)/(√nα)}
we have
P[1
T
T∑i=1
‖Aix‖ > 3s|x|] ≤ C0e−T (α2ns2−ln 2).
We want this true for 3s = C2
1√Tα
, for every x in a 1/2-net on the sphere.
Such a net has cardinality less than 5n. Then successive approximation will
guarantee that inequality (9) be true for all x ∈ Sn−1. The probability that
we get for this is greater than 1 − 5ne−T (nC2/(36T )−ln 2), which for large C is
high, at least in the case where T ≤ C ′n. If T is much larger than n the
term ln 2 interferes, and so we have to use the stronger form of Theorem 1
avoiding this term, which we indicated in the remark following Theorem 1
and also in the proof of the theorem.
In the second case where T ≥ 2( 1αs0
)2 we can no longer take s = C 1√Tα
but only s = s0. The probability is greater than 1 − 5ne−T (α2s0n−ln 2), and
from the assumption on T this probability is exponentially close to 1. So, we
10
get inequality (5). We remark that although we wrote a universal constant
C, the proof shows that this constant is not large at all and can be chosen
to be, say, 5 (and in some cases close to 1). ¤
3 A Local Proposition
In this section we describe an analogue of Milman’s local diameter reduc-
tion theorem, namely the theorem for orthogonal projections. The proof of
Dvoretzky’s Theorem in [Mi1] implies that a random projection of a con-
vex body K in Rn of diameter d = d(K) into a subspace of dimension
k∗ = c(M∗d
)2, is an approximate euclidean ball of radius M∗ = M∗(K). The
fact that this k∗ is indeed the correct formula for the dimension in which a
projection is an approximate euclidean ball, and not just a lower estimate,
was pointed out in [MiS].
It was then observed by Milman that for any dimension k > k∗, when one
projects the body K into a k-dimensional subspace, its diameter decreases
by a factor around√k/n. For a detailed explanation of this fact and more of
the history see [Mi4], Section 2.3.1. Thus, there is only one type of behavior
of the diameter of a convex body under projections, it decreases like√k/n
as long as k is larger than the critical value k∗, and then it stabilizes on the
value M∗. (Note that here we know exact behavior, not only upper bounds.)
In Proposition 3 below we deal with an abstract family of operators satis-
fying a condition which has nothing to do with the body K but only describes
the way the operators act on individual points. Under this condition, a gen-
eral reduction of diameter holds for all convex bodies. We discuss some
11
examples for such families of operators in Section 5, including the classical
case of orthogonal projections.
Proposition 3 Let {A} be some family of operators A : Rn → Rn with some
probability measure P. Assume that for every x ∈ Rn and for any s > s0
P[A : |Ax| > s|x|] ≤ e−cns2
.
Then there exist universal constants c′, C such that given a convex body
K ⊂ Rn the following holds: If s0 > (M∗(K)d(K)
) then for every j ≥ s20n, with
probability greater than 1− e−c′j on the choice of A we have
AK ⊂ C
√j
nd(K)Dn,
and if s0 ≤ (M∗(K)d(K)
) then with probability greater than 1−e−c′ns20 on the choice
of A we have
AK ⊂ CM∗(K)Dn.
Moreover, the constant c′ appearing in the probability is a function of the
constant C, and by increasing C we can have c′ as big as desired.
To prove Proposition 3 we will use Talagrand’s Majorizing Measures The-
orem in the form of Theorem 4 below (see [Tal]); this type of application of
the Majorizing Measure Theorem was suggested to me by Prof. Keith Ball
and was used in a special case in [Art1].
Theorem 4 (Majorizing Measures) There exists a universal constant C0
such that for any dimension n, for every convex body K ⊂ Rn, there exist
12
families of points B0 ⊂ . . . ⊂ Bm−1 ⊂ Bm ⊂ . . . ⊂ K with cardinality
|Bm| ≤ 22msuch that for every x ∈ K
∞∑m=0
d(x,Bm)√
2m ≤ C0M∗(K)
√n,
where d(x,Bm) denotes the distance of the point x to the mth family, i.e.,
d(x,Bm) = inf{d(x, y) : y ∈ Bm}. (Moreover, these points can be constructed
so that if vm is the closest point to x in Bm, and vm+1 is the closest point to
x in Bm+1, then vm is the closest point to vm+1 in Bm.)
Proof of Proposition 3 We fix m0 = log c1j, with c1 universal to be chosen
later. Each vector x ∈ K we write as
x = vm0(x) +∞∑
m=m0+1
(vm(x)− vm−1(x))
where vm(x) ∈ Bm is the member of Bm closest to x. We denote um(x) =
vm(x)−vm−1(x), so that |um(x)| ≤ 2d(x,Bm−1), and denoting Cm = {um(x) :
x ∈ K}, we have |Cm| ≤ 22m+1. We rewrite
x = vm0+∞∑
m=m0+1
(d(x,Bm−1)
√2m−1
C0M∗(K)√n
|um(x)|2d(x,Bm−1))
) (um(x)
|um(x)|2C0M
∗(K)√n√
2m−1
).
The Majorizing Measures Theorem tells us that the sum of the coefficients
in the left brackets of the infinite sum is less than 1. Therefore, for any linear
operator A
|Ax| ≤ |A(vm0(x))|+ supm0<m<∞
|A(um(x))||um(x)|
2C0M∗(K)
√n√
2m−1.
In other words, AK is contained in a ball of radius
maxv∈Bm0
|Av|+ supm0<m<∞,um∈Cm
|Aum||um|
2C0M∗(K)
√n√
2m−1. (10)
13
We first discuss the case s0 > (M∗(K)d(K)
). We use the assumption in the
statement of the Proposition,
P[A : |Ax| > s|x|] ≤ e−cns2
,
to show that with proability 1− e−c′j we have that
(a) For every v ∈ Bm0 ,
|Av| ≤ C1
√j
n|v|,
and (b) For every m > m0, for every um ∈ Cm,
|Aum| ≤ C1d√
2m−1
M∗(K)√n
√j
n|um|.
This will complete the proof of the case s0 > (M∗(K)d
) with C = C1 + 2C0C1.
For (a), since the probability of the event happening for a specific v is at
least 1 − e−cC21 j, we see that if we have in Bm0 no more than ec1j elements,
for c1 = cC21/2, the probability that for all of them we have this property is
greater than 1 − e−c1j, which is precisely why we chose m0 = log c1j (and
we have thus specified c1). In estimating the probability, we have used the
assumption that C1
√j/n > s0, which is clearly satisfied if C1 > 1.
For (b) to happen we take care of each m separately. For each m the
probability is bounded by
1− 22m+1
P [|Ax| > t
√j
n|x|],
for t = dM∗C1
√2m−1√
n. Since we consider m > m0, we have that 2m−1 ≥ c1j,
and hence we can apply the estimate for the probability, as long as, say,
C1 > 1/c1, getting
1− 22m+1
e−cC212m−1 j
n( d
M∗ )2 .
14
We are assuming that s0 > (M∗(K)d(K)
), so jn> (M∗(K)
d)2. Therefore (if, say,
C1 ≥ 4/√c) we can bound this probability from below by 1 − 2−c22m
, and
c2 can be large provided that C1 is chosen large. Adding up for all m the
probability of failure in (b), and adding also the probability of failure in (a)
as the first summand, we see that
e−c1j +∞∑
m=m0+1
2−c22m ≤ 2−c′j,
which completes the proof in the first case. Clearly by increasing C1 we can
increase c′ as much as required.
In the second case, where s0 < (M∗d
), we take m0 = log(c1ns20), again c1
to be chosen later. We use the assumption
P[A : |Ax| > s|x|] ≤ e−cns2
for s = C1s0 to show that with probability 1− e−cC21ns2
0
(a) For every v ∈ Bm0 ,
|Av| ≤ C1s0|v| ≤ C1M∗(K),
and for s = C1
√2m−1√
nto show
(b) For every m > m0, for every um ∈ Cm,
|Aum| ≤ C1
√2m−1
√n
|um|,
and following (10) this will complete the proof in the second case s0 ≤ (M∗d
).
Calculating the probabilities, using that now√
2m/n >√c1s0, we get
that, when C1 > 1/√c1, the probability is greater than
1− e−cC21s2
0n −∞∑
m=m0+1
22m+1
e−cC212m−1
and for C1 sufficiently large the probability is greater than 1− e−c′ns20 . ¤
15
4 A Global Proposition in the reverse direc-
tion
To have a complete global Dvoretzky type statement, Proposition 2 provides
just one direction, namely it shows that the average of a certain amount
of copies of K is contained inside a euclidean ball of an appropriate size.
Actually, under these conditions nothing stronger can be stated. Proposition
5 below gives the reverse side, namely the containment of a ball. Naturally,
it involves a condition which promises that individual points are not shrunk
“too much” by the operators. It also includes an a-priori assumption of
diameter reduction, which can be obtained for example by using Proposition
2. We remark that the condition in the proposition is about a specific value
ε0 which is (four times) the radius of the ball we want inside our body. If one
knows a more global condition satisfied for different values of ε, for example
a small ball probability estimate such as in [LO], one can sometimes get an
“inner diameter increase” lemma by applying the condition each time to a
different pair (ε, T ). In this sense the proof below is very simple, and so can
be adapted to various initial conditions.
Proposition 5 Let {A} be some family of operators A : Rn → Rn with some
probability measure P. Assume that for some body K and some ε0 > 0 the
following holds: for every x ∈ Rn
P[A : ‖Ax‖K < ε0|x|] ≤ (F (ε0))n,
where (F (ε0))n < 1/4. Assume, further, that for some R and some T , with
16
probability p we have
1
T
T∑i=1
‖Aix‖K ≤ R|x|.
Then, if T satisfies T > ln(2 + 16R/ε0)/ ln(1/(2F (ε0))) we have with proba-
bility greater than 1− 2−n − p that
ε0
4|x| ≤ 1
T
T∑i=1
‖Aix‖K ,
where Ai are chosen independently according to the distribution P.
Again we may reformulate the above in geometric form, namely assume
that the family of operators A : Rn → Rn satisfies for some body K and some
ε0 > 0 and F (ε0) < (1/4)1/n, that for every x ∈ Rn one has P[A : ‖A∗x‖∗K <
ε0|x|] ≤ (F (ε0))n, and that for some R and some T , with probability p one has
1T
∑Ti=1AiK ⊂ RDn. Then, if T satisfies T > ln(2+16R/ε0)/ ln(1/(2F (ε0))),
one has with probability greater than 1− 2−n − p that ε0
4Dn ≤ 1
T
∑Ti=1AiK.
Proof of Proposition 5 We will show that for an ε0
4Rnet on the sphere, where
R is the upper bound we are assuming, the inequality holds with ε0/2. Then
by the triangle inequality we will have for every x ∈ Sn−1
1
T
T∑i=1
‖Aix‖K ≥ ε0/4.
The net has cardinality less than (1+8R/ε0)n. From Chernoff, the probability
that more than half of the numbers ‖Aix‖K will be greater than ε0 is greater
than 1 − (2(F (ε0))n)T . The probability that this is true for every point in
the net is greater than
1− (1 + 8R/ε0)n(2F (ε0)
n)T .
17
We now write the condition on T which promises that this quantity is greater
than 1− 2−n and the proof is complete. ¤
5 Some Examples
1. Orthogonal Projections For an integer 1 ≤ k ≤ n let {Pk} be
the family of orthogonal projections onto k-dimensional subspaces of Rn,
endowed with the normalized Haar measure. It is well known, and was
shown for example (with precise estimates on c1, c2 below) in [Art1], that for
s > c1
√kn
P[Pk : |Pkx| > s|x|] ≤ e−c2ns2
.
Proposition 3 then implies Milman’s diameter reduction theorem for random
projections. (And by duality, one sided estimates for Dvoretzky’s Theorem
for sections). Namely, for projection onto a random subspace of dimension k
greater than k∗ = c(M∗d
)2n the diameter decreases like√
kn, and for dimension
lower than this the diameter is around M∗. The other side of Dvoretzky’s
Theorem, namely that a projection onto a subspace of dimension k∗ is already
with high probability isomorphic to a euclidean ball of radius M∗ is not
included in this statement.
2. Sign-Projections For an integer 1 ≤ k ≤ n let {Sk} be the family
of k-dimensional sign-projections, defined as follows: for k sign-vectors εi ∈{−1, 1}n, i = 1, . . . , k let Sk(x) =
∑ki=1〈x, εi/
√n〉ei ∈ Rk. We consider the
uniform measure on this set, namely each εi is chosen with respect to the
uniform measure on the n-dimensional discrete cube. It is not difficult to
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show that for s > c1
√kn
P[Sk : |Skx| > s|x|] ≤ e−c2ns2
.
This is explained for example in [Art2], where in the same spirit as of the
proof of Proposition 3 it was shown that also for this family of operators a
diameter reduction statement holds. In this case, however, the statement is
not sharp, namely for certain bodies, such as B(`n1 ), the decrease of diameter
continues beyond the value k∗. For a more detailed discussion see [Art2].
3. Orthogonal Rotations Consider the family O(n) of orthogonal ro-
tations in Rn, endowed with the normalized Haar measure. Proposition 2
and Proposition 5 together give us the famous global version of Dvoretzky’s
Theorem, namely that the average of ( dM∗ )2 random rotations of a body K
is isomorphic to a euclidean ball of radius M∗. This theorem first appeared
in [BLM]. Indeed, the estimates that we use are the famous concentration of
measure estimate for t > 0
P[x : |‖x‖∗K −M∗| > tM∗|x|] ≤√π/2e−(M∗/d)2t2n/2.
(Where for one side we use this with one specific t = 1/2 say, and apply
Proposition 5, and for the other side we use the tail estimate for t > 1 and
Proposition 2).
4. ψ2-bodies
A convex body T of volume 1 is called a “ψ2-body” if there exists a
constant A such that for each θ ∈ Sn−1 we have that the random variable
Xθ = 〈y, θ〉 (where y is random in T with respect to the volume distribu-
tion) is ψ2 with constant less than A, that is, E(e(Xθ/A)2) ≤ 2. (When the
expectation is with respect to the volume distribution in T )
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We can define another projection-type operator as follows: Consider k
random points yi inside this convex body T (random with respect to the vol-
ume distribution). Define Gk(x) = 1√n
∑ki=1〈x, yi〉ei. That is, a projection-
type operator from Rn into Rk.
To apply Proposition 3, we need to find s0 such that for every x ∈ Rn
and for any s > s0
P[Gk : |Gkx| > s|x|] ≤ e−cns2
.
To do this we use the ψ2 behavior in the standard Markov inequality scheme
P[|Gkx| > s|x|] = P[k∑
i=1
|〈x, yi〉|2 > s2n|x|] ≤
E(e(X/A)2)ke−s2n/A2 ≤ e−n(s2/A2−(k/n) ln 2) ≤ e−k(s2/2A2)
for s > A√
ln 2/2√k/n.
Following Proposition 3 we get exactly the same behavior as in the case
of orthogonal projections and of sign-projections. Notice that we have two
different convex bodies involved, one which helps us define a random opera-
tor, and another whose diameter is reduced by applying this operator. The
first is a ψ2-body, the second is arbitrary.
The case of the same definition of an operator but when the body T with
which we define the random operator is general and not necessarily ψ2 is
different, and is discussed in the next section.
6 Some further continuations
We want to describe two extensions of the above propositions, which, joint
together, are relevant to the example indicated at the end of the above sec-
20
tion.
The first extension of the local Proposition 3 is to the case of ψp behavior,
or more precisely to the case where the tails are not as good, subgaussian,
as in the statement of the proposition. To describe the result we need to
introduce the parameter γp(K), 1 ≤ p ≤ 2, associated to a convex body K.
This parameter was introduced by M. Talagrand in his Majorizing Measures
theory, and is by now a widely used geometric parameter. Define
γp(K) = inf supx∈K
∑m≥0
d(x,Bm)2m/p
where the infimum is taken over all families Bm ⊂ K, m = 0, 1, . . . with
|Bm| ≤ 22m. For p = 2 we have in effect already used the parameter γ2(K)
because γ2(K) ≈ √nM∗(K) (this is exactly Talagrand’s Majorizing Measures
Theorem, one side of which is theorem 4). For motivations for this definition,
computation of γp for certain bodies, and many applications see [Tal].
Using this definition and the method as in the proof of Proposition 3 one
gets the following
Proposition 6 Let {A} be some family of operators A : Rn → Rn with some
probability measure P. Assume that for some 1 ≤ p ≤ 2, for every x ∈ Rn
and for any s > s0
P[A : |Ax| > s|x|] ≤ e−cnsp
.
Then there exist universal constants c′, C such that given a convex body K ⊂Rn the following holds: If s0 > ( γp(K)
d(K)n1/p ) then for every j ≥ sp0n, with
probability greater than 1− e−c′j on the choice of A we have
AK ⊂ C(j
n)1/pd(K)Dn,
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and if s0 ≤ ( γp(K)
d(K)n1/p ) then with probability greater than 1 − e−c′nsp0 on the
choice of A we have
AK ⊂ Cγp(K)
n1/pDn.
Moreover, the constant c′ appearing in the probability is a function of the
constant C, and by increasing C we can have c′ as big as desired.
The second extension we describe is to the case where instead of estimat-
ing the `2-norm of Ax, we consider its size with respect to some other norm.
The most important case to consider is the `p-norm, because if we build a
projection-type operator where the different coordinates are ψp and not ψ2
(for a general body this will be ψ1), then to get an estimate for their sum
we have to sum p-powers of them and not squares (which may have terrible
tails). We discuss the general case of measuring the size of Ax by an arbi-
trary norm ‖ · ‖ with unit ball say B. If our assumption is that the images
shrink in the norm ‖ · ‖, we get that the image of K is inside appropriate
copies of B. More precisely
Proposition 7 Let {A} be some family of operators A : Rn → Rn with
some probability measure P. Let ‖ · ‖ be some abstract norm, with unit ball
B. Assume that for every x ∈ Rn and for any s > s0
P[A : ‖Ax‖ > s|x|] ≤ e−cns2
.
Then there exist universal constants c′, C such that given a convex body
K ⊂ Rn the following holds: If s0 > (M∗(K)d(K)
) then for every j ≥ s20n, with
probability greater than 1− e−c′j on the choice of A we have
AK ⊂ C
√j
nd(K)B,
22
and if s0 ≤ (M∗(K)d(K)
) then with probability greater than 1−e−c′ns20 on the choice
of A we have
AK ⊂ CM∗(K)B.
Moreover, the constant c′ appearing in the probability is a function of the
constant C, and by increasing C we can have c′ as big as desired.
One can similarly combine the two above propositions. The proofs of
these two propositions are similar to the proofs given in this note. Not to
overload this paper technically, we will publish them, together with some
more applications, such as example 4 of Section 5 with T a general convex
body, elsewhere.
References
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[Mi3] V. D. Milman, A note on a low M∗-estimate Geometry of Banach
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24
Shiri Artstein-Avidan,
Department of Mathematics,
Princeton University,
Fine Hall, Washington Road,
Princeton NJ 08544-1000 USA
Email: [email protected]
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